This set of conditions resembles (8) in Theorem 1' of [1]} and Condition (C.2) of [2]}, which essentially requires that all candidate models are misspecified to a non-trivial extent, such that their mean squared errors are not too small. Thus, it precludes any scenario in which the correct models are included in the set of candidate models. To better understand this condition, consider a situation where the set of candidate models includes correct models. In this case, if we denote \(s_0\)  as the index of a correct model, then \(\mu _i=g_{(s_0)}({\bf x}_{(s_0),i}^{\rm T}\beta ^\ast _{(s_0)})\) , and
\(\xi _n&=\inf _{{\bf w}\in {\cal W}}\left\Vert {\mu }^\ast ({\bf w})-{\mu }\right\Vert ^2\le \left\Vert {\bf K}_{(s_0)}({\beta }_{(s_0)}^\ast ){\bf y}-{\mu }\right\Vert ^2=\sum _{i=1}^{n}\left\lbrace \widehat{g}_{(s_0)}({\bf x}_{{(s)},i}^{\rm T}\beta ^\ast _{(s_0)})-\mu _i\right\rbrace ^2\\&=n\left\lbrace O_P(h^2_{s_0}+n^{-1/2}h^{-1/2}_{s_0})\right\rbrace ^2=O_P\left(nh^4_{s_0}+h_{s_0}^{-1}\right),\) 
